Definition df-struct 15697

Description: Define a structure with components in 𝑀...𝑁. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
df-struct	⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

Distinct variable group:   𝑥,𝑓

Detailed syntax breakdown of Definition df-struct

Step	Hyp	Ref	Expression
1		cstr 15691	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1474	. . . . 5 class 𝑥
4		cle 9954	. . . . . 6 class ≤
5		cn 10897	. . . . . . 7 class ℕ
6	5, 5	cxp 5036	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3539	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 1977	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1474	. . . . . 6 class 𝑓
11		c0 3874	. . . . . . 7 class ∅
12	11	csn 4125	. . . . . 6 class {∅}
13	10, 12	cdif 3537	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 5798	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5038	. . . . 5 class dom 𝑓
16		cfz 12197	. . . . . 6 class ...
17	3, 16	cfv 5804	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3540	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1031	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 4642	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1475	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  15703  isstruct2  15704